Lannes' t functor on injective unstable modules and harish-chandra restriction

In the 1980's, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes' T functor on the summands L P := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation theory. We show that T(L P) $\sim$ = L P $\oplus$ H * V 1 $\otimes$ L $\delta$(P) , defining a functor $\delta$ from F p [M n (F p)]-projectives to F p [M n--1 (F p)]-projectives. We relate this new functor $\delta$ to classical constructions in the representation theory of the general linear groups.